Applications Of Derivatives

I'm going to use Induction here.

CASE I (x = 2): $\frac{e^{2}}{2} \ge e \Rightarrow (\frac{e}{2})e \ge e \Rightarrow \frac{e}{2} \ge 1$ , which is true.

CASE II (assume x = k is true): $\frac{e^{k}}{k} \ge e;$

CASE III (verify x = k+1 is true):

$\frac{e^{k}}{k} \cdot e \ge e \cdot e \Rightarrow \frac{e^{k+1}}{k} \ge e^2;$

or $\frac{e^{k+1}}{k} \cdot \frac{k+1}{k+1} \ge e^2;$

or $\frac{e^{k+1}}{k+1} \ge (\frac{k}{k+1}) \cdot e^2;$

or $\frac{e^{k+1}}{k+1} \ge [(1 - \frac{1}{k+1})e] \cdot e$

As $k \rightarrow \infty$ , the quantity $(1 - \frac{1}{k+1}) \cdot e$ approaches $e$ $\Rightarrow \frac{e^{k+1}}{k+1} \ge [(1 - \frac{1}{k+1})e] \cdot e \ge e$ holds true for $x = k+1$ .

$\mathbb{Q}. \mathbb{E}. \mathbb{D}.$